5806 ieee transactions on signal processing...

5806 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 12, DECEMBER 2007

Information Retrieval and Processingin Sensor Networks: DeterministicScheduling Versus Random Access

Min Dong, Member, IEEE, Lang Tong, Fellow, IEEE, and Brian M. Sadler, Fellow, IEEE

Abstract—We investigate the effect of medium-access control(MAC) used in information retrieval by a mobile access point (AP)on information processing in large-scale sensor network, wheresensors are unreliable and subject to outage. We focus on a 1-Dsensor network and assume that the location information is avail-able locally at each sensor and unavailable to the AP. For a fixedcollection interval, two types of MAC schemes are considered: thedeterministic scheduling, which collects data from predeterminedsensors locations, and random access, which collects data fromrandom locations. We compare the signal estimation performanceof the two MACs, using the expected maximum distortion asthe performance measure. For large sensor networks with fixeddensity, we show that there is a critical threshold on the sensoroutage probability out: For out

(1+ (1)), where isthe throughput of the random access protocol, the deterministicscheduling provides better reconstruction performance. However,for out

(1+ (1)), the performance degradation frommissing data samples due to sensor outage does not justify the ef-fort of scheduling; simple random access outperforms the optimalscheduling.

Index Terms—Estimation, information retrieval, random access,random field, sampling, scheduling, sensor network.

I. INTRODUCTION

FOR many applications, a sensor network operates in threephases. In the first phase, sensors take measurements that

form a snapshot of the signal field at a particular time. The mea-surements are stored locally. The second phase is informationretrieval in which data are collected from individual sensors.The last phase is information processing in which data from sen-sors are processed centrally with a specific performance metric.

An appropriate network architecture for such applications isSensor Networks with Mobile Access (SENMA) [1]. Shownin Fig. 1, SENMA has two types of nodes: low-power low-

Manuscript received June 15, 2006; revised February 12, 2007. The as-sociate editor coordinating the review of this manuscript and approvingit for publication was Dr. Mounir Ghogho. This work was supported inpart by the Army Research Laboratory CTA on Communication and Net-works under Grant DAAD19-01-2-0011, the National Science Foundationunder Contract CNS-0435190, and the Army Research Office under GrantARO-W911NF-06-1-0346.

M. Dong is currently with Corporate Research & Development, QualcommInc., San Diego, CA 92121 USA (e-mail: [email protected]).

L. Tong is with the School of Electrical and Computer Engineering, CornellUniversity, Ithaca, NY 14853 USA (e-mail: [email protected]).

B. M. Sadler is with the Army Research Laboratory, Adelphi, MD 20783USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2007.897911

Fig. 1. A 1-D reachback sensor network with a mobile AP.

complexity sensors randomly deployed in large number and afew powerful mobile access points that communicate with sen-sors. Sensors communicate to the mobile access points (APs)directly. The use of mobile APs enable data collections fromspecific areas of the network, either by scheduling or by randomaccess.

We focus on the latter two phases in SENMA: informationretrieval and processing. Specifically, we examine the effect ofmedium-access control (MAC) for information retrieval on in-formation processing in a network in which sensors are unreli-able, possibly with low duty cycle, and subject to outage. In suchcases, it is not possible for mobile access points to collect datafrom all sensors. We consider two types of MAC: deterministicscheduling that schedules transmissions from specific sensor lo-cations, and random access that collects packets randomly fromthe sensor field.

For information processing, packets collected by mobile APsform a sampled signal field, either randomly or deterministi-cally with a specific pattern. Two factors affect the performanceof information processing: the sampling pattern and the MACthroughput. The former tells us how the signal field is sampled,and the latter provides the amount of samples we can obtainduring a collection time using a specific MAC. While it mayappear obvious that collecting data from optimally chosen loca-tions using (centralized) deterministic scheduling gives betterperformance, there are several nontrivial practical complica-tions. For networks with finite sensor density, the schedulingscheme must have a finite scheduling resolution. By this wemean that, because the probability that a randomly deployedsensor exists at a particular location is zero, the deterministicscheduling protocol must be modified to schedule, not a sensorat a particular location, but sensor(s) in the neighborhood of a

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DONG et al.: INFORMATION RETRIEVAL AND PROCESSING IN SENSOR NETWORKS 5807

location. Even with such a modification, there is still non-zeroprobability that the scheduled transmission location is void ofsensors, or the batteries of those scheduled sensors have runout at the time of data collection.

The possibility of sensor outage brings the question ofwhether a deterministic MAC is sufficiently robust to practicalimperfections. In this paper, we compare deterministic sched-uling and random access for the application of reconstructinga signal field with Poisson distributed sensors of finite density.To make the problem more tractable, we study the performancein a 1-D signal field, which provides insight into the two-di-mensional problem. A minimum mean-square error (MMSE)estimator is used for signal reconstruction, and the performancemetric is the expected maximum mean-square error (MSE) ofthe entire signal field. A key parameter is the sensor outageprobability —a function of sensor density, sensor dutycycle, and scheduling resolution—which gives the probabilitythat there is no active sensor within the scheduled resolutioninterval. We show that, for large networks, there is an outageprobability threshold beyond which the deterministic centralscheduling is inferior to distributed random access.

While the literature on sensor networks grows rapidly, littleattention has been paid to the interaction between MAC andsignal reconstruction. We approach this problem by makingthe connection of MAC to sampling and parametrize a MACscheme into a particular sampling scheme. Specifically, wetreat the deterministic scheduling to random access as periodicsampling with missing samples to random sampling. However,because the sampling dictated by the two MAC schemes comesfrom the specific sensor network application, the periodic andrandom sampling in our problem possess their own uniquecharacteristics, and thus are different from the traditionalsampling problem set up [2]–[8]. Therefore, the literature onthe studies of sampling does not provide an answer to ourproblem. The problem of current setting is considered in [9]for the case when the sensor density is infinite without sensoroutage. Using an asymptotic analysis, we show that the optimaldeterministic scheduling always gives better performance thanrandom access, although the performance gain depends on thelevel of measurement signal-to-noise ratio (SNR); the gainin the high-SNR regime is substantial and diminishes at lowSNR. The intuition for the latter is that, in the low-SNR regime,the reconstruction performance is dominated by noise ratherthan by scheduling. In this paper, we generalize that analysisto the case when the sensor density is finite. Moreover, wealso bring the effect of MAC throughput (the probability ofsuccessful transmission) in random access on the performanceinto the comparison. Besides the practical significance of sucha generalization, our results require the notion of sensor outage,drawing the connection of sensor outage in disjoint regions withthe classical problem of head runs in coin tossing [10] and theevaluation of asymptotic extreme statistics of head runs [11].

A. Related Work

In terms of MAC design problem for sensor networks, it hasattracted a growing interest. Many MAC protocols have beenproposed aiming to the special needs and requirements, such

as system throughput and energy efficiency, for both ad hocsensor networks [12]–[14] and reachback sensor networks [15],[16]. In terms of sampling, there have been extensive studiesin the literature. Many studies and performance analysis on theinterpolation methods based on a periodic or random samplescheme have been conducted. See [3] and [4] and referencestherein. In terms of periodic sampling, the effect of jittering, ormissing samples have been studied [6], [17]. Interpolation, se-quential estimation, or prediction using random sampling hasalso been studied [3], [6], [18], [19]. In terms of the comparisonof different type of sampling, the comparisons of periodic sam-pling with Poisson sampling, and time-jittering effect on peri-odic sampling have been considered for some special cases [3],[17], [20]. Despite these results, the discussion on the compar-ison of different sampling schemes is still very limited. Infinitedata samples on a time (or spatial) line are assumed in mostcases. For finite-window samples, the explicit expression is dif-ficult to find and, often, either bounds on the reconstruction errorare derived or numerical comparisons are sorted [3]. For the ap-plication of sampling in sensor networks, in [21], the authorsdiscuss the tradeoff between frequent (equal-spaced) samplingwith 1-bit information per sample and less frequent samplingwith -bit information per sample in a one dimensional signalfield.

B. Organization

The rest of this paper is organized as follows. In Section II, weintroduce the source and sensor network models. In Section III,we explain the deterministic and random access MAC weconsider for data retrieval, followed by the description of theestimator and distortion measure for signal reconstruction. InSection IV, we obtain the expression for the reconstructiondistortion under both MAC schemes for a fixed collectiontime. We then provide the asymptotic performance comparisonunder the two MAC schemes in Section V, where we obtainthe asymptotic behavior of distortion ratio and the thresholdwhich determines the relative performance of the deterministicscheduling and random access. Simulation results are presentedin Section VI. Finally, we conclude in Section VII.

II. SYSTEM MODEL

A. Source Model

We consider a 1-D field of length . The signal of interestin at time is denoted by , where . We assume thatthe spatial dynamic of is a 1-D homogeneous OrnsteinUhlenbeck field [23] governed by the following linear stochasticdifferential equation:

(1)

where and are known, and the process isa standard Brownian motion. The signal ,where , is the stationary solution of (1). Thisprocess is often used to model physical phenomenon such as


temperature, vibration, etc. It is both Gaussian and Markovian[23, Ch. 2]. Given , we have

(2)

Being homogeneous in , has the autocorrelation

(3)

which is only a function of distance between the two pointsand .

B. Sensor Network Model

Consider a network where sensors are randomly deployed in. We assume that the distribution of sensors in forms a 1-D

homogeneous spatial Poisson field with local density sensors/unit area. In particular, in an interval of size , the number ofsensors is a Poisson random variable with probability

(4)

and mean . Given , the sensors areindependent and identically distributed (i.i.d.) in this intervalwith uniform distribution. Finally, the number of sensors in anytwo disjoint intervals are independent.

After deployment, the sensors obtain their location informa-tion through some positioning method [24]–[26]. At time , allsensors measure their local signals and form a snapshot of thesignal field. The measurement of a sensor at location is givenby

(5)

where is spatially i.i.d. zero mean white Gaussian mea-surement noise with variance and is independent of .Note that we consider information retrieval and processing onlybased on the measurement data at time . Thus, we drop the timeindex for brevity in the following presentation.

Each sensor then stores its local measurement, along withtheir sensor location information, in the form of a packet forcollection.

III. INFORMATION RETRIEVAL AND PROCESSING

A. MAC for Information Retrieval

When the mobile AP is ready for data collection, it retrievespackets from sensors within a fixed collection time using a givenMAC scheme. Assuming slotted transmissions in a collisionchannel,1 we consider two types of MAC schemes for data re-trieval: deterministic scheduling and random access.

We assume that the AP has no knowledge of the sensor lo-cations before data collection, and only each sensor knows its

1A packet is correctly received only if no other users attempt transmission.

own location information. If the AP had sensor location infor-mation, it could exploit it. However, this implies a great deal ofoverhead and additional networking burden. Sensor deploymentis often random, and additional sensors may be added at a latertime; each sensor’s operational condition may also be uncertain(sensors may be out of operation, or in different duty cycles).All these hinder a priori knowledge of sensor location informa-tion at the AP, and a large overhead burden occurs if the APwere to acquire this before data collection, especially in a largesensor field. Therefore, we focus on the case where the sensorsknow their individual locations, but location information is notrequired at the AP.

1) Deterministic Scheduler : A deterministic schedulingcollects data from predetermined locations. Given a collectiontime of slots, one particular scheme is to preselect sen-sors to transmit. If the AP has the knowledge of sensor locations,there is an optimal selection of sensors to transmit. Withoutthe knowledge of sensor location, however, it is nontrivial toselect a good set of sensors at the initial stage before col-lection. Somehow, the AP will need to travel across the sensorfield and learn where the sensors are, and the overhead of thisinitial learning process can be significant. Furthermore, batteryexpiration, addition of new sensors, and different sensor dutycycles can all make the eligible sensor set time varying. In addi-tion, some sensors may not have new data to transmit. That alsomakes the deterministic selection of sensors for data collec-tion inappropriate.

Instead, we consider a deterministic scheme that schedulessensors from the intervals centered at equally spaced lo-cations to access the channel. Ideally, if sensors exist at theseequally spaced locations, such a scheduling is optimal for thedistortion measure we consider in Section III-C [9]. However,when there is no sensor at a scheduled location, deterministicschedulings result in missed samples. For randomly distributedsensors, the probability that a sensor exists at a particular loca-tion is zero. Thus, we need to consider scheduling with finiteresolution.

Specifically, the AP obtains the squally spaced locationsbased on the field length and the collection time slots.It then travels through the field for data collection, where thescheduler enables a resolution interval of length centeredat each desired location, and the AP collects one packet fromeach interval.2 If there are no active sensors in a resolutioninterval, we say a sensor outage occurs. Let be the proba-bility of sensor outage. For the target , the smallest intervallength that enables should satisfy

(6)

In other words, the resolution interval is determined by andas

(7)

2This can be implemented through carrier sensing by letting the sensor closestto the center of the interval to transmit [22], or using a control channel.


Notice that the maximum number of disjoint intervalsthat can enable is

(8)

In summary, for given target sensor outage probabilityand network density , the deterministic scheduler first setsthe resolution interval length according to (7). Forgiven slots of collection time ( ), it next enables

intervals of length centered at the equally spacedlocations. Then, the AP schedules one packet transmission (datasample) from each interval. Note that the AP can use targetand choose resolution interval based on , or it may choose toonly use as the resolution interval, andis the resulting outage probability. We choose the former way inthe following presentation, which separates the notion of collec-tion interval from the collection time to allow more flex-ible design choices under practical constraints.3 We note thatboth finite and asymptotic analytical results derived below arethe same under either approach.

2) Random Access : In the random access, in contrast,sensors contend to access the channel with equal priority. Weassume that packets from sensors have equal reception prob-ability,4 and therefore the origins of received packets in arerandom. In terms of the sample pattern, these received data sam-ples can be viewed as if they are results of the AP randomly sam-pling the sensor measurement data in . Specifically, the mobileAP activates sensors in an area for transmissions with a givenrandom access protocol. Depending on the application, the areaenabled can be either the whole field or randomly chosen sub-regions of . For an enabled collection area in the network, let

be the MAC throughput (probability of successful transmis-sion per slot) for the area with activated sensors. We assumethat exists.

B. Viewing MAC as Sampling Scheme

As we have mentioned above, a MAC scheme determines aspecific sampling scheme: It dictates the data sample amountand underling sampling pattern. Therefore, in terms of the effecton signal reconstruction, we can view a MAC scheme as a sam-pling scheme. In particular, for the two types of MAC schemeswe consider, the deterministic scheduler can be viewed as pe-riodic sampling. With the existence of sensor outage in a reso-lution interval, we can view the effect of scheduling as periodicsampling with missing samples. For random access, on the otherhand, the resulting sampling pattern is random with uniform dis-

3Using intervalD=M may not be always possible or desirable; We considerthe resolution interval to be a design choice. Specifically, the access point usu-ally has to be confined to an angle of vision. If the antenna of the access pointhas the fixed beamwidth, then it is appropriate to fix the retrieval interval size,instead of a function of M . At the sensor field, if the segment D=M is large,sensors may not be able to hear each other. This increases the chance of colli-sion and resulting loss of data in that interval. If the sensor density is high, itis not necessary to enable the wholeD=M segment, but rather a small intervalmay be sufficient.

4In this work, we assumed all nodes have equal reception probabilities (de-sired for reconstruction purpose). For large spatial network, this can be realizedby dividing the field into smaller regions and the mobile AP enables one regionat a time.

Fig. 2. Example of the locations of received packets.

tribution. Therefore, we can view the retrieval result as randomsampling (in both pattern and amount) with uniform distribu-tion. Note that the sampling induced by the MAC scheme is dif-ferent from ones in traditional sampling problems: The size ofthe sensor field is finite; for random access, besides the samplinglocation is random, the number of samples in a given collectiontime is limited and random with the mean equal to wherethe expectation is taken over , which is Poisson distributed.In this paper, we are interested in the comparison of the per-formance of the two sampling schemes dictated by the MACschemes within the same collection time, and their asymptoticbehavior of the MAC performance in a large network as the re-trieval time becomes large.

C. Information Processing and Performance Measure

Assume that the AP retrieves packets from with a MACscheme for slots, and packets in are received. Notethat is random for both and (due to sensor outage).We reconstruct the original signal based on these received datasamples, and measure the signal reconstruction performance.

To avoid the boundary effect for signal reconstruction, we as-sume that there is a sensor deployed at each of the two bound-aries of , and we are able to obtain their measurements. Thelocations of all the received packets5 are denoted by

. We denote the corresponding order sta-tistics of by . Also, wedenote and . Fig. 2 shows a realizationof sampling locations of the signal field in .

We estimate at location using its two immediateneighbor samples by the MMSE smoothing. That is, for

,

(9)

In the noiseless measurement case where , theabove estimator is in fact optimal in terms of MSE. In otherwords

(10)

The reason is that, due to the Markov property of in (1),the signal , where , is only a functionof and . When measurement noise is present,the estimator in (9) is suboptimal. Nonetheless, its simple struc-ture and easy computation lends itself as an attractive practicalestimator.

5For convenience,M only denotes the number of packets not from the twoboundary sensors of A.


Let the measurement SNR be SNR . Given ,we use the maximum mean square estimation error in as theperformance measure for the reconstruction distortion in a givenrealization of received samples:

SNR (11)

Note that maximum MSE is often used to capture the worst caseperformance. Specific to the application of environmental mon-itoring by sensor network, information from each local area isneeded in order to observe any abnormal behavior or activity.For information accuracy, it requires that the error of the recon-structed field at each location should be less than a threshold.The maximum MSE precisely provides this information, andtherefore is used here as the performance measure.

A MAC scheme specifies where and how data packets shouldbe transmitted, in other words, how the signal field

is sampled; it not only determines the number of sam-ples received, but also specifies the distribution of sample loca-tions . The performance of a MAC scheme is then mea-sured by the expected value of signal distortion metric in (11)under a specific sampling scheme dictated by . In other words,the expected maximum distortion of signal reconstruction with

distinct received packets is then given by

SNR SNR (12)

where the expectation is taken over for . Theexpected maximum distortion of a MAC during the collectiontime slots is then given by

SNR SNR (13)

where the expectation is taken over .Our objective is to compare the signal reconstruction distor-

tion SNR of the two MAC schemes at different sensoroutage condition , when each scheme is used to retrievedata for a fixed collection time.

IV. SIGNAL RECONSTRUCTION DISTORTION AND MAC

In this section, we obtain the average maximum distortionSNR of the two schemes and .

Given received data samples, the maximum distortionSNR in (11) can be found by comparing the maximum

distortion of in each interval , for. Because the estimate in (9) is only a func-

tion of the two immediate neighbor samples of , we have

SNR

(14)

Note that the signal and the measurement noise areindependent and Gaussian. Given , and

are jointly Gaussian. Therefore, the MMSE estimator

in (9) is linear. The expressions for and its MSE areobtained by

(15)

where, and . With (3), we can show

that the maximum MSE of , for , isonly a function of the distance between and

SNR

(16)

where , for . The super-script of denotes the number of received distinct data samples.The maximum distortion SNR in (14) is therefore de-termined only by the maximum of distances betweenany two adjacent data samples

SNR SNR

(17)where , which is a monotone in-creasing function of . The average maximum distortionwith received samples in (12) is now rewritten as

SNR SNR (18)

A. Random Access

As we have mentioned in Section III, for the random-ac-cess MAC , given received samples, their locations

are i.i.d. random variables with uniform distri-bution . Consequently, the maximum sample distance

is random. The cumulative distribution offor can be calculated from the distribution

of the order statistics . The joint density ofis given by [27]

ifotherwise.

(19)Using the transformation of random variables, we have the jointdistribution of sample distances given by

if , andotherwise.

(20)


Since , we have

(21)

where is the indicator function: for, and otherwise. By calculating the above, the

expression of is given by (22), shown at the bottomof the page. The distribution of is plotted in Fig. 3 forseveral values of as examples.

Combining (18) and (22), and applyingfor , the expected maximum distortion for with

packets is obtained by

SNR SNR

SNR

SNR (23)

where is such that SNR , and SNR is thederivative of SNR given by

SNR

Finally, the expected maximum distortion of in (13) is ob-tained by

SNR SNR

SNR (24)

B. Deterministic Scheduler

For the deterministic scheduler with target sensor outageprobability , it enables intervals of length cen-tered at , for and collects one samplefrom each interval. The actual number of received samples maybe less than due to sensor outage. In the later presentation,

Fig. 3. Distribution of d under � (D = 1): (a) CDF of d ; (b) PDFof d .

we use instead of for simplicity. The readers shouldkeep in mind of the relationship of with and . From (13)and (18), we have

SNR SNR (25)

where now is the maximum sample distance in a realiza-tion of the received data samples, which is determined by the

if

if

ifif .

(22)


largest number of consecutive intervals that sensor outageoccurs. Then, we have

(26)The expected maximum distortion SNR with istherefore bounded by

SNR

SNR

SNR (27)

where

SNR

(28)

To calculate the bounds, we need to find the cumulative dis-tribution of denoted by . It is given by

(29)

where is the number of realizations in which exactintervals have sensor outage, but no more than of these areconsecutive.

Since sensors are Poisson distributed, for disjoint resolutionintervals, the occurrence of sensor outage in each of them areindependent with probability . This can be viewed as a se-quence of a biased coin tossing, where heads appear with .Then, the distribution of is equivalent to that of the longesthead run (consecutive heads) in a sequence of biased cointosses. The coefficient therefore can be viewed as thenumber of sequences of length in which exactly headsoccur, but no more than of these occur consecutively. Thelongest run in a sequence of coin tossing has been studied ex-tensively [10], [11]. It is shown in [10] that can be cal-culated by the following recursion:

ifif

if .(30)

Therefore, the upper and lower bounds on SNR can becalculated using (27), (29), and (30).

V. SCHEDULING VERSUS RANDOM ACCESS

In this section, we perform an analysis for large sensor net-works. The average number of sensors in the network, denotedas , is given by . We are interested in the comparison

of the asymptotic performance of the two schemes when islarge.

Under the mechanisms of deterministic scheduler andrandom access described in Section III-A, we compare thethe signal reconstruction distortion of the two schemes. Thefollowing theorem describes the asymptotic behavior of recon-struction distortion SNR when becomes large.

Theorem 1: Let be given. Suppose that (1) as; (2) as , the reconstruction

distortions under and are given by6

SNRSNR

(31)

where .

SNRSNR

(32)

where .Proof: See Appendix I.

Theorem 1 shows us how reconstruction distortion scaleswith as the network grows large (either density, or both sizeand density). Note that the condition on the growth of networkpopulation in Theorem 1 allows the network size growbut below rate with increased density . One specialcase of this condition is the case where the size of the network

is fixed and the density grows. Note from the first term of(31) and (32) that, there exists a lower bound on the distortion,we denote it by SNR :

SNRSNR

(33)

From (17), we see that this bound corresponds to the case whenwe let the density of the sensor network go to infinity (thus thereexists a sensor at any location), and collect all the measure-ment of the field. The distortion in the second term of (31) and(32) corresponds to the distortion due to the particular samplingscheme dictated by deterministic scheduler and random ac-cess . It is this term that represents the difference of signalreconstruction under different MAC schemes.

The distortion expressions in (31) and (32) show that in largesensor network, as we use more and more time slots to col-lect the data samples, the reconstruction distortion decreases tothe lower bound SNR . The decreasing rate depends on theunderlying sampling scheme dictated by the MAC scheme, aswell as the signal spatial correlation. From the second terms in

6A function f(x) is O(g(x)) if lim f(x)=g(x) = c where c < 1is constant; f(x) is o(g(x)) if lim f(x)=g(x) = 0, in particular, wheng(x) = 1, f(x) is o(1) if lim f(x) = 0.


Fig. 4. Distortion decay ratio r(M; SNR).

(31) and (32), we see that, for , the decreasing rate is at least, and for , the rate is at least .

Given the asymptotic expression in Theorem 1, we now com-pare the performance of the two schemes and . Note thatwe could directly compare the distortion SNR of the twoschemes. But as we mentioned above, the term representing thedistortion due to the specific MAC scheme used is the secondterm in (31) and (32). Therefore, we will only compare thesecond term, i.e., the excess distortion

SNR SNR

We will see in Theorem 2 that comparing the excess distortionprovides a clear expression for the threshold on accordingto which the relative performance of and changes. Forconvenience, we define the ratio of the excess distortion ofto that of as

SNR SNR

SNR SNR(34)

Fig. 4 illustrates the ratio of the two MAC schemes. Therelative performance of and , therefore, depends on thevalue of comparing with 1: If , is better than ;otherwise, outperforms . The following theorem showsthe condition for (or 1) in a large sensor network.

Theorem 2: Let be given. Suppose that 1) asand 2) as . Then, for

and for .Proof: By (31) and (32) in Theorem 1, we have the ratio

as

(35)

We see that the second term goes to 1 as grows. Therefore,the ratio is determined by the first leading term. Comparing itto 1, we have the threshold on .

From (35), it is clear that the relative performance of anddepends critically on . As we see, although the deter-

ministic scheduling is designed for an optimal retrieval pattern,its performance is less robust in the sense that it suffers frommissing data samples due to sensor outage. Theorem 2 gives usthe transition of the relative performance of the two strategies.

Fig. 5. Favored region of the two MAC schemes.

For , the gain of scheduling transmissionwith desired pattern is evident, although there exists a mild per-formance loss because of sensor outage. As we expected, thedeterministic scheduler provides a better reconstruction perfor-mance. In [9], we have considered the case when the sensor den-sity goes to infinity, and therefore there is no sensor outage. Wehave shown that, in fact, the excess distortion ratio grows as

. When , however, The-orem 1 shows that the performance loss of due to missingdata samples does not justify the effort of scheduling the de-sired retrieval pattern, and random access outperforms the op-timal scheduling.

Note that there are two factors of a MAC scheme affecting thesignal reconstruction: the sampling pattern and the throughputof MAC. The former determines where the samples are from,and the latter determines how many samples the AP can ob-tain. Because of sensor outage, the resulting sampling patternby using degrades the performance due to missing data sam-ples. For the random access , on the other hand, the number ofsamples is limited by the throughput. The thresholdon for the relative performance clearly indicates the abovetwo factors: the condition is equivalent to

, i.e., we should set the length of the resolutioninterval so that the expected number of sensors in the resolutioninterval is greater than . Fig. 5 depicts the favoredregion of each MAC scheme. The two regions are divided bythe curve . For fixed , the resolution interval

needs to be larger than to let perform better.On the other hand, if we keep the of fixed and vary , thenfor lower than a certain value, the expected number of sensorsin is too small, and it is better to perform random access. Ourresult implies that if the average number of received data sam-pling in random access is more than that in deterministic sched-uler, then we may just let sensors perform random access insteadof careful planning for desired retrieval pattern. This shows thatalthough the deterministic scheduler can be optimally designed,it relies on the network parameters and is therefore less robustto the imperfect knowledge of the network conditions.

VI. SIMULATIONS

We now numerically compare the reconstruction perfor-mance of and for various scenarios. The expectedmaximum distortion of and is calculated using (24) and(27), respectively.


Fig. 6. Signal field S(x) (D = 500 m, f = 0.02 m, � = 1).

Fig. 7. Expected maximum distortion ��E(M; SNR) versus M (� =5 sensors/m). (a) P = 0:5; (b) P = 0:75.

A. Signal Field and System Setup

We consider a field with 500 m. For the signal ofinterest , an example of the Ornstein Uhlenbeck field isshown in Fig. 6. We set and 0.05 m (i.e., for

1 m, ). We set the sensor density5 sensors/m and randomly deploy Poisson distributed sen-

sors in . For the random access, we choose a simple ALOHAwith delay-first transmission without feedback, where each of

Fig. 8. ��E(M;SNR) versus P (� = 5 sensors/m, M = 300).

the activated sensors transmits its packet with probability .In this case, the throughput, i.e., the probability of a successfultransmission, is and

.

B. Distortion versus Collection Time

We plot in Fig. 7 the expected maximum distortionSNR of and with various for SNR 0,

20 dB, respectively. Fig. 7(a) and (b) are the plots for prob-ability of sensor outage and ,respectively. We observe that as increases, on average thereceived data samples at the AP increases, and SNR ofboth and decreases. However, we see that the relativereconstruction performance of and depends on thedifferent value of we set. For the relatively high weset, the reconstruction performance of the random access isuniformly better than that of the deterministic scheduler .

C. Distortion Versus Sensor Outage Probability

To see the transition of the relative performance of and, we plot SNR of and with for

various at SNR 0, 20 dB in Fig. 8. Note that sinceis the design parameter for , the performance of does notchange for different . We observe that clearly there exist athreshold on approximately at 0.67 , such thatthe performance of is worse than for all greater thanthis value, and better for less than this value.

We also compare the ratio of excess distortion obtained inour simulations with the theoretical expression of in (35) ofTheorem 2, where the we use the coefficients in (36) of Lemma1 and (57) of Lemma 3 to replace the ’s in (35). Fig. 9 showsthe curves of versus for SNR 20 dB and .We see that the simulation matches the result in Theorem 2.

D. Distortion Versus SNR

In Fig. 10, we plot the expected maximum distortionSNR of and with various SNR for .

We plot SNR of at 0.5, 0.75, where the


Fig. 9. Distortion ratio r versus P (� = 5 sensors/m, M = 300, SNR =20 dB).

Fig. 10. ��E(M; SNR) versus SNR (� = 5 sensors/m, M = 300).

performance of is better than that of at 0.5and worse at 0.75. For each case, we see that as SNRbecomes large, the performance gap between and in-creases. As expected, when noise level is high, the distortionis mainly dominated by the measurement noise, therefore theperformance gap is small. When SNR becomes higher, the ef-fect on the reconstruction distortion due to using different MACscheme becomes more significant, and the performance gap be-comes larger.

E. Distortion versus Signal Spatial Correlation

Finally, we compare the performance under various signalspatial correlation levels. From (3), the signal correlationcoefficient is given by , where . InFig. 11(a) and (b), we plot SNR of and with var-ious for SNR 20 and 0 dB, respectively. We setand . For ranges from 0.02 to 0.8, the correlationcoefficient ranges from 0.98 (high correlation) to 0.45

Fig. 11. ��E(M;SNR) versus correlation coefficient a (� = 5 sensors/m, M =300). (a) SNR = 20 dB; (b) SNR = 0 dB.

(low correlation). As expected, the lower spatial correlation ,the higher distortion. For the performance gap at different corre-lation level, from Theorem 2, we see that is not in the leadingterm of the expression for . Therefore, for relatively large,the spatial correlation does not affects the ratio . For high SNR,where the limiting distortion SNR is very small, the ex-cess distortion dominants SNR . Thus, the performancegap (in decibels) is relatively unchanged. For low SNR, how-ever, SNR is relatively large. For high spatial correlation,

SNR is dominated by the limiting distortion SNR .Thus, the performance gap (in decibels) when is high is lesssignificant than that when is low.

VII. CONCLUSION

In this paper, we investigate how MAC for information re-trieval affects information processing. Specifically, we considerthe performance of signal field reconstruction based on re-ceived data. Data retrieval with different MAC schemes resultsin distinct sampling pattern of the field, and throughput of MAC


determines the number of samples received by the AP. Ouranalysis demonstrates that, for MAC design in sensor networks,a cross-layer approach that integrates application and MACshould be considered. In particular, affecting the choices ofMAC are sensor density and reconstruction performance, bothdictated by the source characteristics and network parameters.

Using the maximum reconstruction distortion as our perfor-mance measure, we examined two MAC strategies: the deter-ministic scheduling designed to collect data from equally spacedlocations and the random access MAC. We have shown that, al-though the scheduling is designed according to the optimal re-trieval pattern, its performance suffers from the possibility ofsensor outage, which results in missing data samples and, there-fore, is less robust to the uncertainty of the network parameters.Specifically, the relative reconstruction performance is shownto depend critically on the outage probability of sensors,which determines the average number of active sensors in a reso-lution interval. Our analysis shows that, for networks with largesensor population, whether a deterministic scheduler is betterthan random access depends on whether the critical threshold is

. The former is better only if .Based on this, the favored region of each MAC is shown. Thetwo regions are divided by , where isthe expected number of sensors in the resolution interval , and

corresponds to . Thisindicates that if the average number of samples obtained usingthe deterministic scheduler is less than that of random access ina given collection time, then one may just let sensors performrandom access.

Although our results provide insight of the relation of MACand application performance in sensor network, many problemsremain for further study. In this paper, the analysis and com-parison of different MAC schemes for signal reconstruction islimited to the 1-D sensor field. Even though the results provideinsight into the 2-D network, detailed studies of the interactionof MAC and signal reconstruction performance in the 2-D fieldare still of interest. The extension to the 2-D case is nontrivialas the signal modeling and the reconstruction metric expres-sion become much more complex. In order to make the problemmore tractable for analysis, some level of simplified approxi-mation for the 2-D models and signal estimation method maybe assumed. For example, the field can be quantized into dis-crete grids, and a Gauss–Markov field on the lattice, often usedfor 2-D modeling, can be applied to study our problem. Also,in this study, we focus on the MMSE signal reconstruction forthe random signal model considered in (1). A more generalizedrandom signal model can be considered for further detailed ex-amination. The performance in many other applications, suchas signal prediction, spectral estimation, is also of great interestfor investigation.

APPENDIX IPROOF OF THEOREM 1

A. Asymptotic Distortion of Deterministic Scheduler

To calculate the expected maximum distortion SNRof , we first need to find the statistics of , the largest

number of consecutive intervals that have sensor outage. ForPoisson-distributed sensors, the sensor outage in each disjointinterval of length is i.i.d. with probability . The problemof finding essentially is equivalent to that of finding thelongest run of consecutive heads for a sequence of biasedcoin tossing, where the probability of head is . Using theextreme value theory of head runs [11], we show that pos-sesses an almost sure behavior. For convenience, let

The result is shown in the following lemma.Lemma 1: Let be defined as before. Then

(36)Therefore

a.s. (37)

Proof: Theorem 3 and Theorem 4 of [11] provide almostsure behavior of . We state these theorems below.

Theorem 3: Let be a nondecreasing sequence of in-tegers. Then, is 0 or 1 asfinite or infinite.

Proof: See [11].Theorem 4: Let be a nondecreasing sequence of

integers with and. Then,

is 0 or 1, as determined by whetheris finite or in-

finite.Proof: See [11].

Let . By Theorem 3

(38)

This implies

(39)

On the other hand, for , it satisfiesthe two conditions in Theorem 4:

(40)

(41)

By Theorem 4

(42)


This implies

(43)

Combining (39) and (43), (36) and (37) follows.Using Lemma 1, , such that , the lower

bound of the expected maximum distortion of in (27) canbe rewritten as

SNR

(44)

where since , we have. For function , as ,

using Taylor’s Theorem, we have

(45)

If the growth rate for is such that as, then we can apply (45) to (44), and we obtain the expected

maximum distortion as

SNR

SNR

SNR

(46)

The upper bound in (27) can be similarly derived and has thesame order as in (44). Therefore, we have the asymptotic ex-pression in (31).

B. Asymptotic Distortion of Random Access

1) Asymptotic Expression of SNR for Large :Now, we obtain the distortion of the random access , given

received packets from distinct sensors. From (17), the max-imum distortion of is a function of

(47)

where , is the order statistics of i.i.d. uni-formly distributed random variables. The property of isgiven in the following lemma.

Lemma 2: Let be i.i.d. unit exponential random variables,and , for . Then, the joint distributionof is the same as that of the order sta-tistics from i.i.d. random variables with uniform distribution

, i.e.,

ifotherwise.

(48)Proof: The joint distribution of is given by

(49)

where , . Note that .

For any , let . By transforma-tion of random variables, we then have the joint density of

given by

(50)Notice that has a gamma density

It follows that

(51)

Therefore, from (50) and (51), we have the joint density

(52)

By Lemma 2, then the joint distribution of theorder statistics is the same as that of

. Therefore, the jointdistribution of is the same as that of

, where. Thus, the maximum distance can be represented

by

(53)

Therefore, we have

(54)


By the Strong Law of Large Numbers, we have

a.s. (55)

Let . From the asymptotic theory of ex-treme statistics, we have the almost sure asymptotic property of

as follows.Lemma 3 (Logarithmic Growth Rate of [28, p. 262], [29,

p. 215]): Let be the maximum of i.i.d. unit exponentialrandom variables. Then

(56)Combining (54)–(56), we have

(57)Thus, similarly as in (44)–(46), if as

, for large enough of received samples, theexpected maximum distortion of can be obtained as

SNRSNR

(58)

We now obtain the limiting distribution of SNR .Without loss of generality, we consider that the AP activates thesensors in the whole field for transmissions. Note that, given

, the throughput is just the probability of a successfultransmission. Thus, is binomial distributed with

(59)

Because , for a given , , s.t. for all ,. By the Chebyshev inequality, for fixed , we have

(60)

(61)

Now we bound SNR . Note that SNR de-creases with . We have

SNR

SNR

SNR

SNR (62)Therefore

SNR

SNR

SNR

SNR

Note that

(63)

Similarly

(64)

Following SNR in (58) for large , we have the lim-iting distortion as

SNR SNR

SNR(65)

2) Lower Bound on SNR SNR : We nowlower bound SNR SNR for large and .We have

(66)

The lower bound on the excess distortion is given by

SNR SNR

SNR SNR

SNR

SNR

SNR

SNR (67)

Since , for any , , s.t. for all

(68)

Also we have

SNR

SNR (69)


Since SNR is a decreasing function of , we have

SNR SNR

(70)

Combining (67) and (70), we have

SNR SNR SNR

The tail probability of can be again bounded by the Cheby-shev inequality. Then, , , s.t. for all

(71)Since the number of sensors in is Poisson distributed, for any

(72)

Thus, for any

s.t. for all

(73)

Thus, for and

SNR SNR

SNR

SNR (74)

From (58), for large , we have

SNR

(75)

Therefore, by (74), for large and , we have thelower bound

SNR SNR

(76)

where .3) Upper Bound on SNR SNR : For the

upper bound, we have

SNR SNR

SNR

SNR

SNR (77)

Since SNR is a decreasing function of , for the firstterm, we have

SNR

SNR (78)For the second term, we have

SNR

SNR (79)As in the derivation of the lower bound, , s.t. for all

(80)

Therefore

(81)

(82)

for some , where the bound on the second term in (81) issimilar to that in (60). Therefore

SNRSNR

(83)Therefore, combining (78) and (83), , for

, the excess distortion in (77) is upper bounded by

SNR SNR SNR(84)

for some . From (75), for large and , wehave

SNR SNR

(85)

Combining the lower and upper bounds in (76) and (84), wehave, for large and

SNR SNR

(86)

With (65), the asymptotic expression in (32) follows.


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[1] L. Tong, Q. Zhao, and S. Adireddy, “Sensor networks with mobileagents,” presented at the 2003 Military Communications Int. Symp.,Boston, MA, Oct. 2003.

[2] C. E. Shannon, “Communication in the presence of noise,” Proc. IRE,pp. 10–21, Jan. 1949.

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[8] D. Klamer and E. Masry, “Polynomial interpolation of randomly sam-pled bandlimited functions and processes,” SIAM J. Appl. Math., vol.42, pp. 1004–1019, 1982.

[9] M. Dong, L. Tong, and B. Sadler, “Impact of data retrieval pattern onhomogeneous signal field reconstruction in dense sensor networks,”IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4352–4364, Nov.2006.

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[22] Z. Yang, M. Dong, L. Tong, and B. M. Sadler, “MAC protocols foroptimal information retrieval pattern in sensor networks with mobileaccess,” EURASIP J. Wireless Commun. Netw., no. 4, pp. 493–504,2005.

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Min Dong (S’00–M’05) received the B.Eng. degreefrom Tsinghua University, Beijing, China, in 1998and the Ph.D. degree in electrical engineering with aminor in applied mathematics from Cornell Univer-sity, Ithaca, NY, in 2004.

She is currently with Corporate Research and De-velopment, Qualcomm, Inc., San Diego, CA, whereshe has been actively involved in the orthogonal fre-quency-division multiple-access (OFDMA) systemdesign and performance analysis. Her researchinterests are in the general areas of communications,

signal processing, and mobile networks.Dr. Dong received the 2004 IEEE Signal Processing Society Best Paper

Award.

Lang Tong (S’87–M’91–SM’01–F’05) receivedthe B.E. degree from Tsinghua University, Beijing,China, in 1985, and the M.S. and Ph.D. degreesin electrical engineering from the University ofNotre Dame, Notre Dame, IN, in 1987 and 1991,respectively.

He was a Postdoctoral Research Affiliate at the In-formation Systems Laboratory, Stanford University,Stanford, CA, in 1991. Prior to joining Cornell Uni-versity, he was on faculty at West Virginia Universityand the University of Connecticut. He was also the

2001 Cor Wit Visiting Professor at the Delft University of Technology, Deflt,The Netherlands. He is currently the Irwin and Joan Jacobs Professor in Engi-neering at Cornell University, Ithaca, NY. His research is in the general areaof statistical signal processing, wireless communications and networking, andinformation theory.

Dr. Tong received the 1993 Outstanding Young Author Award from the IEEECircuits and Systems Society, the 2004 Best Paper Award (with M. Dong) fromthe IEEE Signal Processing Society, and the 2004 Leonard G. Abraham PrizePaper Award from the IEEE Communications Society (with P. Venkitasubra-maniam and S. Adireddy). He is also a coauthor of five student paper awards.He received the Young Investigator Award from the Office of Naval Research.He has served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL

PROCESSING, the IEEE TRANSACTIONS ON INFORMATION THEORY, and IEEESIGNAL PROCESSING LETTERS.

Brian M. Sadler (M’90–SM’02–F’06) receivedthe B.S. and M.S. degrees from the University ofMaryland, College Park, and the Ph.D. degree fromthe University of Virginia, Charlottesville, all inelectrical engineering.

He is a Senior Research Scientist at the ArmyResearch Laboratory (ARL), Adelphi, MD. He wasa Lecturer at the University of Maryland and hasbeen lecturing at The Johns Hopkins University,Baltimore, MD, since 1994 on statistical signal pro-cessing and communications. His research interests

include signal processing for mobile wireless and ultra-wideband systems,sensor signal processing and networking, and associated security issues.

Dr. Sadler is currently an Associate Editor for the IEEE SIGNAL PROCESSING

LETTERS and the IEEE TRANSACTIONS ON SIGNAL PROCESSING and has beena Guest Editor for several journals, including IEEE JOURNAL OF SELECTED

TOPICS IN SIGNAL PROCESSING, the IEEE JOURNAL OF SELECTED AREAS IN

COMMUNICATIONS, and the IEEE Signal Processing Magazine. He is a memberof the IEEE Signal Processing Society Sensor Array and Multi-Channel Tech-nical Committee, and received a Best Paper Award (with R. Kozick) from theSignal Processing Society in 2006.

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